Quasiconformal solutions to elliptic partial differential equations

Abstract

In this paper, we assume that \(G\) and \(\Omega\) are two Jordan domains in \(\mathbb{R}^n\) with \(\mathcal{C}^2\) boundaries, where \(n\ge 2\), and prove that every quasiconformal mapping \(f\in\mathcal{W}^{2,1+\epsilon}_{\mathrm{loc}}\) of \(G\) onto \(\Omega\), satisfying the elliptic partial differential inequality \(|L_ A[f]|\lesssim (\|Df\|^2+|g|)\), with \(g\in\mathcal{L}^p(G)\), where \(p>n\), is Lipschitz continuous. The result is sharp since for \(p=n\), the mapping \(f\) is not necessarily Lipschitz continuous. This extends several results for harmonic quasiconformal mappings.